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LES of Turbulent Flows: Lecture 2 Supplement(ME EN 7960-003)

Prof. Rob StollDepartment of Mechanical Engineering

University of Utah

Fall 2014

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Fourier Transforms are a common tool in fluid dynamics (see Pope, Appendix D-G, Stull handouts online)

Some uses:

• Analysis of turbulent flow

• Numerical simulations of N-S equations

• Analysis of numerical schemes (modified wavenumbers)

• consider a periodic function f(x) (could also be f(t)) on a domain of length 2π

• The Fourier representation of this function (or a general signal) is:

- where k is the wavenumber (frequency if f(t))

- are the Fourier coefficients which in general are complex

Fourier Transforms

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Fourier Transforms• why pick eikx ?

- answer: orthogonality

- a big advantage of orthogonality is independence between Fourier modes

- eix is independent of ei2x just like we have with Cartesian coordinates

where i,j,k are all independent of each other.

• what are we actually doing?

• recall from Euler’s formula that

• the Fourier transform decomposes a signal (space or time) into sine and cosine wave components of different amplitudes and wave numbers (or frequencies).

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Fourier Transforms• Fourier Transform example (from Stull, 88 see example: FourierTransDemo.m)

Real cosinecomponent

Real sinecomponent

Sum ofwaves

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Fourier Transforms• The Fourier representation given by is a representation of a series as a

function of sine and cosine waves. It takes f(x) and transforms it into wave space

• Fourier Transform pair: consider a periodic function on a domain of 2π

• The forwards transform moves us into Fourier (or wave) space and the backwards transform moves us from wave space back to real space.

• An alternative form of the Fourier transform (using Euler’s) is:

Where ak and bk are now the real and imaginary components of fk, respectively

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Locality in real and wave space• It is important to note that something that is non local in physical (real) space can

be very local in Fourier space and something that is local in physical space can be non local in Fourier space.

• Example 1:

• Example 2:

½

21 k

fk

21 k

fk1/2π

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1. If f(x) is real then:

2. Parseval’s Theorem:

3. The Fourier representation is the best possible representation for f(x) in the sense that the error:

Fourier Transform Properties

(Complex conjugate)

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Discrete Fourier Transform• Consider the periodic function fj on the domain 0 ≤ x ≤ L

• Discrete Fourier Representation:

• we have: known: fj at N points

unknown: at k values (N of them)• Using discrete orthogonality:

j=0 j=1 j=3 j=Nwith xj=jh and h = L/N

Periodicity implies that f0=fN

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Discrete Fourier Transform• Discrete Fourier Transform (DFT) example and more explanation found in

handouts section of website under Stull_DFT.pdf or Pope appendix F and in example: FourierTransDemo.m

• Implementation of DFT by brute force O(N2) operations• In practice we almost always use a Fast Fourier Transform (FFT) O(N*log2N)• Almost all FFT routines (e.g., Matlab, FFTW, Intel, Numerical Recipes, etc.) save

their data with the following format:

positions xj

wavenumbers

Nyquist

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Fourier Transform ApplicationsAutocorrelation:• We can use the discrete Fourier Transform to speed up the autocorrelation

calculation (or in general any cross-correlation with a lag).

-Discretely this is O(N2) operations

-if we express Rff as a Fourier series

-how can we interpret this??-In physical space

correlation with itself

magnitude of the Fourier coefficients

total contribution to the variance

energy spectral density

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Fourier Transform ApplicationsEnergy Spectrum: (power spectrum, energy spectral density)

• If we look at specific k values from our autocorrelation function we have:

where E(k) is the energy spectral density

• Typically (when written as) E(k) we mean the contribution to the turbulent kinetic energy (tke) = ½(u2+v2+w2) and we would say that E(k) is the contribution to tke for motions of the scale (or size) k . For a single velocity component in one direction we would write E11(k1).

• This means that the energy spectrum and

the autocorrelation function form a Fourier

Transform pair (see Pope for details)

The square of the Fourier coefficients is the contribution to the variance by fluctuations of scale k (wavenumber or equivalently frequency

E(k)

k

Example energy spectrum

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Sampling TheoremBand-Limited function: a function where

Theorem: If f(x) is band limited, i.e., , then f(x) is completely represented by its values on a discrete grid, where n is an integer

(-∞< n < ∞) and kc is called the Nyquist frequency.

Implication:• If we have with a domain of 2π:

If the number of points is ≥ 2kc then the discrete Fourier Transform=exact solution

e.g., for f(x)=cos(6x) we need N ≥ 12 points to represent the function exactly

E(k)

k-kkc-kc

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Sampling Theorem• What if f(x) is not band-limited?

• or f(x) is band limited but sampled at a rate < kc, for example f(x)=cos(6x) with 8 points.

• Result: Aliasing contamination of resolved energy by energy outside of the resolved scales.

k

E(k)

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Aliasing• Consider:

=1 (integer function of 2π)

k

E(k)

What does this mean for spectra?

exact

aliased

What is actually happening to E(k)?

E(k)

k-k kc-kc Negative wavenumbers folded to here

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Aliasing Example

Fourier coefficients (all real since only cosine)

k-k 1 2 3 4 5 6-1-2-3-4-5-6

Consider N=8 kc = N/2 = 4

k-k 1 2 3 4 5 6-1-2-3-4-5-6

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Aliased points

Consider a function:

Aliasing if N

For more on Fourier Transforms see Pope Ch. 6, online handout from Stull or Press et al., Ch 12-13.